<?xml version="1.0" encoding="ascii"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
          "DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
  <title>weblogolib.logomath</title>
  <link rel="stylesheet" href="epydoc.css" type="text/css" />
  <script type="text/javascript" src="epydoc.js"></script>
</head>

<body bgcolor="white" text="black" link="blue" vlink="#204080"
      alink="#204080">
<!-- ==================== NAVIGATION BAR ==================== -->
<table class="navbar" border="0" width="100%" cellpadding="0"
       bgcolor="#a0c0ff" cellspacing="0">
  <tr valign="middle">
  <!-- Home link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="weblogolib-module.html">Home</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Tree link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="module-tree.html">Trees</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Index link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="identifier-index.html">Indices</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Help link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="help.html">Help</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Project homepage -->
      <th class="navbar" align="right" width="100%">
        <table border="0" cellpadding="0" cellspacing="0">
          <tr><th class="navbar" align="center"
            ><a class="navbar" target="_top" href="http://code.google.com/p/weblogo/">WebLogo</a></th>
          </tr></table></th>
  </tr>
</table>
<table width="100%" cellpadding="0" cellspacing="0">
  <tr valign="top">
    <td width="100%">
      <span class="breadcrumbs">
        <a href="weblogolib-module.html">Package&nbsp;weblogolib</a> ::
        Module&nbsp;logomath
      </span>
    </td>
    <td>
      <table cellpadding="0" cellspacing="0">
        <!-- hide/show private -->
      </table>
    </td>
  </tr>
</table>
<h1 class="epydoc">Source Code for <a href="weblogolib.logomath-module.html">Module weblogolib.logomath</a></h1>
<pre class="py-src">
<a name="L1"></a><tt class="py-lineno">  1</tt>  <tt class="py-line"><tt class="py-comment">#!/usr/bin/env python</tt> </tt>
<a name="L2"></a><tt class="py-lineno">  2</tt>  <tt class="py-line"><tt class="py-comment"></tt> </tt>
<a name="L3"></a><tt class="py-lineno">  3</tt>  <tt class="py-line"><tt class="py-comment"># -------------------------------- WebLogo --------------------------------</tt> </tt>
<a name="L4"></a><tt class="py-lineno">  4</tt>  <tt class="py-line"><tt class="py-comment"></tt> </tt>
<a name="L5"></a><tt class="py-lineno">  5</tt>  <tt class="py-line"><tt class="py-comment">#  Copyright (c) 2003-2004 The Regents of the University of California.</tt> </tt>
<a name="L6"></a><tt class="py-lineno">  6</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  Copyright (c) 2005 Gavin E. Crooks</tt> </tt>
<a name="L7"></a><tt class="py-lineno">  7</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  Copyright (c) 2006-2011, The Regents of the University of California, through </tt> </tt>
<a name="L8"></a><tt class="py-lineno">  8</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  Lawrence Berkeley National Laboratory (subject to receipt of any required</tt> </tt>
<a name="L9"></a><tt class="py-lineno">  9</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  approvals from the U.S. Dept. of Energy).  All rights reserved.</tt> </tt>
<a name="L10"></a><tt class="py-lineno"> 10</tt>  <tt class="py-line"><tt class="py-comment"></tt> </tt>
<a name="L11"></a><tt class="py-lineno"> 11</tt>  <tt class="py-line"><tt class="py-comment">#  This software is distributed under the new BSD Open Source License.</tt> </tt>
<a name="L12"></a><tt class="py-lineno"> 12</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  &lt;http://www.opensource.org/licenses/bsd-license.html&gt;</tt> </tt>
<a name="L13"></a><tt class="py-lineno"> 13</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#</tt> </tt>
<a name="L14"></a><tt class="py-lineno"> 14</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  Redistribution and use in source and binary forms, with or without </tt> </tt>
<a name="L15"></a><tt class="py-lineno"> 15</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  modification, are permitted provided that the following conditions are met: </tt> </tt>
<a name="L16"></a><tt class="py-lineno"> 16</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#</tt> </tt>
<a name="L17"></a><tt class="py-lineno"> 17</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  (1) Redistributions of source code must retain the above copyright notice, </tt> </tt>
<a name="L18"></a><tt class="py-lineno"> 18</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  this list of conditions and the following disclaimer. </tt> </tt>
<a name="L19"></a><tt class="py-lineno"> 19</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#</tt> </tt>
<a name="L20"></a><tt class="py-lineno"> 20</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  (2) Redistributions in binary form must reproduce the above copyright </tt> </tt>
<a name="L21"></a><tt class="py-lineno"> 21</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  notice, this list of conditions and the following disclaimer in the </tt> </tt>
<a name="L22"></a><tt class="py-lineno"> 22</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  documentation and or other materials provided with the distribution. </tt> </tt>
<a name="L23"></a><tt class="py-lineno"> 23</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#</tt> </tt>
<a name="L24"></a><tt class="py-lineno"> 24</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  (3) Neither the name of the University of California, Lawrence Berkeley </tt> </tt>
<a name="L25"></a><tt class="py-lineno"> 25</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  National Laboratory, U.S. Dept. of Energy nor the names of its contributors </tt> </tt>
<a name="L26"></a><tt class="py-lineno"> 26</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  may be used to endorse or promote products derived from this software </tt> </tt>
<a name="L27"></a><tt class="py-lineno"> 27</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  without specific prior written permission. </tt> </tt>
<a name="L28"></a><tt class="py-lineno"> 28</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#</tt> </tt>
<a name="L29"></a><tt class="py-lineno"> 29</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" </tt> </tt>
<a name="L30"></a><tt class="py-lineno"> 30</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE </tt> </tt>
<a name="L31"></a><tt class="py-lineno"> 31</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE </tt> </tt>
<a name="L32"></a><tt class="py-lineno"> 32</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE </tt> </tt>
<a name="L33"></a><tt class="py-lineno"> 33</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR </tt> </tt>
<a name="L34"></a><tt class="py-lineno"> 34</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF </tt> </tt>
<a name="L35"></a><tt class="py-lineno"> 35</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS </tt> </tt>
<a name="L36"></a><tt class="py-lineno"> 36</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN </tt> </tt>
<a name="L37"></a><tt class="py-lineno"> 37</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) </tt> </tt>
<a name="L38"></a><tt class="py-lineno"> 38</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE </tt> </tt>
<a name="L39"></a><tt class="py-lineno"> 39</tt>  <tt class="py-line"><tt class="py-comment"></tt><tt class="py-comment">#  POSSIBILITY OF SUCH DAMAGE. </tt> </tt>
<a name="L40"></a><tt class="py-lineno"> 40</tt>  <tt class="py-line"><tt class="py-comment"></tt> </tt>
<a name="L41"></a><tt class="py-lineno"> 41</tt>  <tt class="py-line"><tt class="py-keyword">from</tt> <tt class="py-name">math</tt> <tt class="py-keyword">import</tt> <tt class="py-name">log</tt><tt class="py-op">,</tt> <tt class="py-name">sqrt</tt><tt class="py-op">,</tt> <tt class="py-name">exp</tt> </tt>
<a name="L42"></a><tt class="py-lineno"> 42</tt>  <tt class="py-line"> </tt>
<a name="L43"></a><tt class="py-lineno"> 43</tt>  <tt class="py-line"><tt class="py-keyword">from</tt> <tt class="py-name">numpy</tt> <tt class="py-keyword">import</tt> <tt class="py-name">array</tt><tt class="py-op">,</tt> <tt class="py-name">asarray</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">,</tt> <tt class="py-name">ones</tt><tt class="py-op">,</tt> <tt class="py-name">zeros</tt><tt class="py-op">,</tt> <tt class="py-name">int32</tt><tt class="py-op">,</tt><tt class="py-name">all</tt><tt class="py-op">,</tt><tt class="py-name">any</tt><tt class="py-op">,</tt> <tt class="py-name">shape</tt> </tt>
<a name="L44"></a><tt class="py-lineno"> 44</tt>  <tt class="py-line"><tt class="py-keyword">import</tt> <tt class="py-name">numpy</tt> <tt class="py-name">as</tt> <tt class="py-name">na</tt> </tt>
<a name="L45"></a><tt class="py-lineno"> 45</tt>  <tt class="py-line"><tt class="py-keyword">from</tt> <tt class="py-name">corebio</tt><tt class="py-op">.</tt><tt class="py-name">moremath</tt> <tt class="py-keyword">import</tt> <tt class="py-op">*</tt> </tt>
<a name="L46"></a><tt class="py-lineno"> 46</tt>  <tt class="py-line"> </tt>
<a name="L47"></a><tt class="py-lineno"> 47</tt>  <tt class="py-line"><tt class="py-keyword">import</tt> <tt class="py-name">random</tt> </tt>
<a name="L48"></a><tt class="py-lineno"> 48</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet"></a><div id="Dirichlet-def"><a name="L49"></a><tt class="py-lineno"> 49</tt> <a class="py-toggle" href="#" id="Dirichlet-toggle" onclick="return toggle('Dirichlet');">-</a><tt class="py-line"><tt class="py-keyword">class</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html">Dirichlet</a><tt class="py-op">(</tt><tt class="py-base-class">object</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="Dirichlet-expanded"><a name="L50"></a><tt class="py-lineno"> 50</tt>  <tt class="py-line">    <tt class="py-docstring">"""The Dirichlet probability distribution. The Dirichlet is a continuous </tt> </tt>
<a name="L51"></a><tt class="py-lineno"> 51</tt>  <tt class="py-line"><tt class="py-docstring">    multivariate probability distribution across non-negative unit length</tt> </tt>
<a name="L52"></a><tt class="py-lineno"> 52</tt>  <tt class="py-line"><tt class="py-docstring">    vectors. In other words, the Dirichlet is a probability distribution of </tt> </tt>
<a name="L53"></a><tt class="py-lineno"> 53</tt>  <tt class="py-line"><tt class="py-docstring">    probability distributions. It is conjugate to the multinomial</tt> </tt>
<a name="L54"></a><tt class="py-lineno"> 54</tt>  <tt class="py-line"><tt class="py-docstring">    distribution and is widely used in Bayesian statistics.</tt> </tt>
<a name="L55"></a><tt class="py-lineno"> 55</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L56"></a><tt class="py-lineno"> 56</tt>  <tt class="py-line"><tt class="py-docstring">    The Dirichlet probability distribution of order K-1 is </tt> </tt>
<a name="L57"></a><tt class="py-lineno"> 57</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L58"></a><tt class="py-lineno"> 58</tt>  <tt class="py-line"><tt class="py-docstring">     p(theta_1,...,theta_K) d theta_1 ... d theta_K = </tt> </tt>
<a name="L59"></a><tt class="py-lineno"> 59</tt>  <tt class="py-line"><tt class="py-docstring">        (1/Z) prod_i=1,K theta_i^{alpha_i - 1} delta(1 -sum_i=1,K theta_i)</tt> </tt>
<a name="L60"></a><tt class="py-lineno"> 60</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L61"></a><tt class="py-lineno"> 61</tt>  <tt class="py-line"><tt class="py-docstring">    The normalization factor Z can be expressed in terms of gamma functions:</tt> </tt>
<a name="L62"></a><tt class="py-lineno"> 62</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L63"></a><tt class="py-lineno"> 63</tt>  <tt class="py-line"><tt class="py-docstring">      Z = {prod_i=1,K Gamma(alpha_i)} / {Gamma( sum_i=1,K alpha_i)}  </tt> </tt>
<a name="L64"></a><tt class="py-lineno"> 64</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L65"></a><tt class="py-lineno"> 65</tt>  <tt class="py-line"><tt class="py-docstring">    The K constants, alpha_1,...,alpha_K, must be positive. The K parameters, </tt> </tt>
<a name="L66"></a><tt class="py-lineno"> 66</tt>  <tt class="py-line"><tt class="py-docstring">    theta_1,...,theta_K are nonnegative and sum to 1.</tt> </tt>
<a name="L67"></a><tt class="py-lineno"> 67</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L68"></a><tt class="py-lineno"> 68</tt>  <tt class="py-line"><tt class="py-docstring">    Status:</tt> </tt>
<a name="L69"></a><tt class="py-lineno"> 69</tt>  <tt class="py-line"><tt class="py-docstring">        Alpha</tt> </tt>
<a name="L70"></a><tt class="py-lineno"> 70</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L71"></a><tt class="py-lineno"> 71</tt>  <tt class="py-line">    <tt class="py-name">__slots__</tt> <tt class="py-op">=</tt> <tt class="py-string">'alpha'</tt><tt class="py-op">,</tt> <tt class="py-string">'_total'</tt><tt class="py-op">,</tt> <tt class="py-string">'_mean'</tt><tt class="py-op">,</tt>  </tt>
<a name="L72"></a><tt class="py-lineno"> 72</tt>  <tt class="py-line">     </tt>
<a name="L73"></a><tt class="py-lineno"> 73</tt>  <tt class="py-line">     </tt>
<a name="L74"></a><tt class="py-lineno"> 74</tt>  <tt class="py-line">     </tt>
<a name="L75"></a><tt class="py-lineno"> 75</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet.__init__"></a><div id="Dirichlet.__init__-def"><a name="L76"></a><tt class="py-lineno"> 76</tt> <a class="py-toggle" href="#" id="Dirichlet.__init__-toggle" onclick="return toggle('Dirichlet.__init__');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#__init__">__init__</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">alpha</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.__init__-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.__init__-expanded"><a name="L77"></a><tt class="py-lineno"> 77</tt>  <tt class="py-line">        <tt class="py-docstring">"""</tt> </tt>
<a name="L78"></a><tt class="py-lineno"> 78</tt>  <tt class="py-line"><tt class="py-docstring">        Args:</tt> </tt>
<a name="L79"></a><tt class="py-lineno"> 79</tt>  <tt class="py-line"><tt class="py-docstring">            - alpha  -- The parameters of the Dirichlet prior distribution.</tt> </tt>
<a name="L80"></a><tt class="py-lineno"> 80</tt>  <tt class="py-line"><tt class="py-docstring">                        A vector of non-negative real numbers.  </tt> </tt>
<a name="L81"></a><tt class="py-lineno"> 81</tt>  <tt class="py-line"><tt class="py-docstring">        """</tt> </tt>
<a name="L82"></a><tt class="py-lineno"> 82</tt>  <tt class="py-line">        <tt class="py-comment"># TODO: Check that alphas are positive</tt> </tt>
<a name="L83"></a><tt class="py-lineno"> 83</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-comment">#TODO : what if alpha's not one dimensional?</tt> </tt>
<a name="L84"></a><tt class="py-lineno"> 84</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">asarray</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L85"></a><tt class="py-lineno"> 85</tt>  <tt class="py-line">         </tt>
<a name="L86"></a><tt class="py-lineno"> 86</tt>  <tt class="py-line">        <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_total</tt> <tt class="py-op">=</tt> <tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> </tt>
<a name="L87"></a><tt class="py-lineno"> 87</tt>  <tt class="py-line">        <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_mean</tt> <tt class="py-op">=</tt> <tt class="py-name">None</tt> </tt>
</div><a name="L88"></a><tt class="py-lineno"> 88</tt>  <tt class="py-line">         </tt>
<a name="L89"></a><tt class="py-lineno"> 89</tt>  <tt class="py-line">         </tt>
<a name="Dirichlet.sample"></a><div id="Dirichlet.sample-def"><a name="L90"></a><tt class="py-lineno"> 90</tt> <a class="py-toggle" href="#" id="Dirichlet.sample-toggle" onclick="return toggle('Dirichlet.sample');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#sample">sample</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.sample-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.sample-expanded"><a name="L91"></a><tt class="py-lineno"> 91</tt>  <tt class="py-line">        <tt class="py-docstring">"""Return a randomly generated probability vector.</tt> </tt>
<a name="L92"></a><tt class="py-lineno"> 92</tt>  <tt class="py-line"><tt class="py-docstring">        </tt> </tt>
<a name="L93"></a><tt class="py-lineno"> 93</tt>  <tt class="py-line"><tt class="py-docstring">        Random samples are generated by sampling K values from gamma</tt> </tt>
<a name="L94"></a><tt class="py-lineno"> 94</tt>  <tt class="py-line"><tt class="py-docstring">        distributions with parameters a=\alpha_i, b=1, and renormalizing. </tt> </tt>
<a name="L95"></a><tt class="py-lineno"> 95</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L96"></a><tt class="py-lineno"> 96</tt>  <tt class="py-line"><tt class="py-docstring">        Ref:</tt> </tt>
<a name="L97"></a><tt class="py-lineno"> 97</tt>  <tt class="py-line"><tt class="py-docstring">            A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).</tt> </tt>
<a name="L98"></a><tt class="py-lineno"> 98</tt>  <tt class="py-line"><tt class="py-docstring">        Authors:</tt> </tt>
<a name="L99"></a><tt class="py-lineno"> 99</tt>  <tt class="py-line"><tt class="py-docstring">            Gavin E. Crooks &lt;gec@compbio.berkeley.edu&gt; (2002)</tt> </tt>
<a name="L100"></a><tt class="py-lineno">100</tt>  <tt class="py-line"><tt class="py-docstring">        """</tt> </tt>
<a name="L101"></a><tt class="py-lineno">101</tt>  <tt class="py-line">        <tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> </tt>
<a name="L102"></a><tt class="py-lineno">102</tt>  <tt class="py-line">        <tt class="py-name">K</tt> <tt class="py-op">=</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> </tt>
<a name="L103"></a><tt class="py-lineno">103</tt>  <tt class="py-line">        <tt class="py-name">theta</tt> <tt class="py-op">=</tt> <tt class="py-name">zeros</tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-name">K</tt><tt class="py-op">,</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L104"></a><tt class="py-lineno">104</tt>  <tt class="py-line"> </tt>
<a name="L105"></a><tt class="py-lineno">105</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">k</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">K</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L106"></a><tt class="py-lineno">106</tt>  <tt class="py-line">            <tt class="py-name">theta</tt><tt class="py-op">[</tt><tt class="py-name">k</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">random</tt><tt class="py-op">.</tt><tt class="py-name">gammavariate</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">k</tt><tt class="py-op">]</tt><tt class="py-op">,</tt> <tt class="py-number">1.0</tt><tt class="py-op">)</tt>  </tt>
<a name="L107"></a><tt class="py-lineno">107</tt>  <tt class="py-line">        <tt class="py-name">theta</tt> <tt class="py-op">/=</tt> <tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">theta</tt><tt class="py-op">)</tt> </tt>
<a name="L108"></a><tt class="py-lineno">108</tt>  <tt class="py-line"> </tt>
<a name="L109"></a><tt class="py-lineno">109</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">theta</tt> </tt>
</div><a name="L110"></a><tt class="py-lineno">110</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet.mean"></a><div id="Dirichlet.mean-def"><a name="L111"></a><tt class="py-lineno">111</tt> <a class="py-toggle" href="#" id="Dirichlet.mean-toggle" onclick="return toggle('Dirichlet.mean');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#mean">mean</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.mean-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.mean-expanded"><a name="L112"></a><tt class="py-lineno">112</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt>  <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_mean</tt> <tt class="py-op">==</tt><tt class="py-name">None</tt><tt class="py-op">:</tt> </tt>
<a name="L113"></a><tt class="py-lineno">113</tt>  <tt class="py-line">            <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_mean</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> <tt class="py-op">/</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_total</tt> </tt>
<a name="L114"></a><tt class="py-lineno">114</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">_mean</tt> </tt>
</div><a name="L115"></a><tt class="py-lineno">115</tt>  <tt class="py-line">     </tt>
<a name="Dirichlet.covariance"></a><div id="Dirichlet.covariance-def"><a name="L116"></a><tt class="py-lineno">116</tt> <a class="py-toggle" href="#" id="Dirichlet.covariance-toggle" onclick="return toggle('Dirichlet.covariance');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#covariance">covariance</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.covariance-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.covariance-expanded"><a name="L117"></a><tt class="py-lineno">117</tt>  <tt class="py-line">        <tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> </tt>
<a name="L118"></a><tt class="py-lineno">118</tt>  <tt class="py-line">        <tt class="py-name">A</tt> <tt class="py-op">=</tt> <tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> </tt>
<a name="L119"></a><tt class="py-lineno">119</tt>  <tt class="py-line">        <tt class="py-comment">#A2 = A * A</tt> </tt>
<a name="L120"></a><tt class="py-lineno">120</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-name">K</tt> <tt class="py-op">=</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> </tt>
<a name="L121"></a><tt class="py-lineno">121</tt>  <tt class="py-line">        <tt class="py-name">cv</tt> <tt class="py-op">=</tt> <tt class="py-name">zeros</tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-name">K</tt><tt class="py-op">,</tt><tt class="py-name">K</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt>  </tt>
<a name="L122"></a><tt class="py-lineno">122</tt>  <tt class="py-line">         </tt>
<a name="L123"></a><tt class="py-lineno">123</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">K</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L124"></a><tt class="py-lineno">124</tt>  <tt class="py-line">            <tt class="py-name">cv</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">,</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-number">1.</tt> <tt class="py-op">-</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt><tt class="py-op">/</tt><tt class="py-name">A</tt><tt class="py-op">)</tt> <tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-name">A</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">1.</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
<a name="L125"></a><tt class="py-lineno">125</tt>  <tt class="py-line">         </tt>
<a name="L126"></a><tt class="py-lineno">126</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">K</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L127"></a><tt class="py-lineno">127</tt>  <tt class="py-line">            <tt class="py-keyword">for</tt> <tt class="py-name">j</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">i</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">,</tt><tt class="py-name">K</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L128"></a><tt class="py-lineno">128</tt>  <tt class="py-line">                <tt class="py-name">v</tt> <tt class="py-op">=</tt> <tt class="py-op">-</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">*</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">j</tt><tt class="py-op">]</tt> <tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-name">A</tt> <tt class="py-op">*</tt> <tt class="py-name">A</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">1.</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
<a name="L129"></a><tt class="py-lineno">129</tt>  <tt class="py-line">                <tt class="py-name">cv</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">,</tt><tt class="py-name">j</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">v</tt> </tt>
<a name="L130"></a><tt class="py-lineno">130</tt>  <tt class="py-line">                <tt class="py-name">cv</tt><tt class="py-op">[</tt><tt class="py-name">j</tt><tt class="py-op">,</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">v</tt> </tt>
<a name="L131"></a><tt class="py-lineno">131</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">cv</tt> </tt>
</div><a name="L132"></a><tt class="py-lineno">132</tt>  <tt class="py-line">         </tt>
<a name="Dirichlet.mean_x"></a><div id="Dirichlet.mean_x-def"><a name="L133"></a><tt class="py-lineno">133</tt> <a class="py-toggle" href="#" id="Dirichlet.mean_x-toggle" onclick="return toggle('Dirichlet.mean_x');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#mean_x">mean_x</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.mean_x-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.mean_x-expanded"><a name="L134"></a><tt class="py-lineno">134</tt>  <tt class="py-line">        <tt class="py-name">x</tt> <tt class="py-op">=</tt> <tt class="py-name">asarray</tt><tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L135"></a><tt class="py-lineno">135</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">shape</tt><tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">)</tt> <tt class="py-op">!=</tt> <tt class="py-name">shape</tt><tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L136"></a><tt class="py-lineno">136</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">"Argument must be same dimension as Dirichlet"</tt><tt class="py-op">)</tt> </tt>
<a name="L137"></a><tt class="py-lineno">137</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">sum</tt><tt class="py-op">(</tt> <tt class="py-name">x</tt> <tt class="py-op">*</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-0" class="py-name" targets="Method weblogolib.logomath.Dirichlet.mean()=weblogolib.logomath.Dirichlet-class.html#mean,Method weblogolib.logomath.Gamma.mean()=weblogolib.logomath.Gamma-class.html#mean"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-0', 'mean', 'link-0');">mean</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt>  </tt>
</div><a name="L138"></a><tt class="py-lineno">138</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet.variance_x"></a><div id="Dirichlet.variance_x-def"><a name="L139"></a><tt class="py-lineno">139</tt> <a class="py-toggle" href="#" id="Dirichlet.variance_x-toggle" onclick="return toggle('Dirichlet.variance_x');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#variance_x">variance_x</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.variance_x-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.variance_x-expanded"><a name="L140"></a><tt class="py-lineno">140</tt>  <tt class="py-line">        <tt class="py-name">x</tt> <tt class="py-op">=</tt> <tt class="py-name">asarray</tt><tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L141"></a><tt class="py-lineno">141</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">shape</tt><tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">)</tt> <tt class="py-op">!=</tt> <tt class="py-name">shape</tt><tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L142"></a><tt class="py-lineno">142</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">"Argument must be same dimension as Dirichlet"</tt><tt class="py-op">)</tt>         </tt>
<a name="L143"></a><tt class="py-lineno">143</tt>  <tt class="py-line">             </tt>
<a name="L144"></a><tt class="py-lineno">144</tt>  <tt class="py-line">        <tt class="py-name">cv</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-1" class="py-name" targets="Method weblogolib.logomath.Dirichlet.covariance()=weblogolib.logomath.Dirichlet-class.html#covariance"><a title="weblogolib.logomath.Dirichlet.covariance" class="py-name" href="#" onclick="return doclink('link-1', 'covariance', 'link-1');">covariance</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L145"></a><tt class="py-lineno">145</tt>  <tt class="py-line">        <tt class="py-name">var</tt> <tt class="py-op">=</tt> <tt class="py-name">na</tt><tt class="py-op">.</tt><tt class="py-name">dot</tt><tt class="py-op">(</tt><tt class="py-name">na</tt><tt class="py-op">.</tt><tt class="py-name">dot</tt><tt class="py-op">(</tt><tt class="py-name">na</tt><tt class="py-op">.</tt><tt class="py-name">transpose</tt><tt class="py-op">(</tt> <tt class="py-name">x</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">cv</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt class="py-name">x</tt><tt class="py-op">)</tt> </tt>
<a name="L146"></a><tt class="py-lineno">146</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">var</tt> </tt>
</div><a name="L147"></a><tt class="py-lineno">147</tt>  <tt class="py-line"> </tt>
<a name="L148"></a><tt class="py-lineno">148</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet.mean_entropy"></a><div id="Dirichlet.mean_entropy-def"><a name="L149"></a><tt class="py-lineno">149</tt> <a class="py-toggle" href="#" id="Dirichlet.mean_entropy-toggle" onclick="return toggle('Dirichlet.mean_entropy');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#mean_entropy">mean_entropy</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.mean_entropy-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.mean_entropy-expanded"><a name="L150"></a><tt class="py-lineno">150</tt>  <tt class="py-line">        <tt class="py-docstring">"""Calculate the average entropy of probabilities sampled</tt> </tt>
<a name="L151"></a><tt class="py-lineno">151</tt>  <tt class="py-line"><tt class="py-docstring">        from this Dirichlet distribution. </tt> </tt>
<a name="L152"></a><tt class="py-lineno">152</tt>  <tt class="py-line"><tt class="py-docstring">        </tt> </tt>
<a name="L153"></a><tt class="py-lineno">153</tt>  <tt class="py-line"><tt class="py-docstring">        Returns:</tt> </tt>
<a name="L154"></a><tt class="py-lineno">154</tt>  <tt class="py-line"><tt class="py-docstring">            The average entropy.</tt> </tt>
<a name="L155"></a><tt class="py-lineno">155</tt>  <tt class="py-line"><tt class="py-docstring">            </tt> </tt>
<a name="L156"></a><tt class="py-lineno">156</tt>  <tt class="py-line"><tt class="py-docstring">        Ref:</tt> </tt>
<a name="L157"></a><tt class="py-lineno">157</tt>  <tt class="py-line"><tt class="py-docstring">            Wolpert &amp; Wolf, PRE 53:6841-6854 (1996) Theorem 7</tt> </tt>
<a name="L158"></a><tt class="py-lineno">158</tt>  <tt class="py-line"><tt class="py-docstring">            (Warning: this paper contains typos.)</tt> </tt>
<a name="L159"></a><tt class="py-lineno">159</tt>  <tt class="py-line"><tt class="py-docstring">        Status:</tt> </tt>
<a name="L160"></a><tt class="py-lineno">160</tt>  <tt class="py-line"><tt class="py-docstring">            Alpha</tt> </tt>
<a name="L161"></a><tt class="py-lineno">161</tt>  <tt class="py-line"><tt class="py-docstring">        Authors:</tt> </tt>
<a name="L162"></a><tt class="py-lineno">162</tt>  <tt class="py-line"><tt class="py-docstring">            GEC 2005</tt> </tt>
<a name="L163"></a><tt class="py-lineno">163</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L164"></a><tt class="py-lineno">164</tt>  <tt class="py-line"><tt class="py-docstring">        """</tt> </tt>
<a name="L165"></a><tt class="py-lineno">165</tt>  <tt class="py-line">        <tt class="py-comment"># TODO: Optimize</tt> </tt>
<a name="L166"></a><tt class="py-lineno">166</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> </tt>
<a name="L167"></a><tt class="py-lineno">167</tt>  <tt class="py-line">        <tt class="py-name">A</tt> <tt class="py-op">=</tt> <tt class="py-name">float</tt><tt class="py-op">(</tt><tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L168"></a><tt class="py-lineno">168</tt>  <tt class="py-line">        <tt class="py-name">ent</tt> <tt class="py-op">=</tt> <tt class="py-number">0.0</tt> </tt>
<a name="L169"></a><tt class="py-lineno">169</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">a</tt> <tt class="py-keyword">in</tt> <tt class="py-name">alpha</tt><tt class="py-op">:</tt> </tt>
<a name="L170"></a><tt class="py-lineno">170</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">a</tt><tt class="py-op">&gt;</tt><tt class="py-number">0</tt> <tt class="py-op">:</tt> <tt class="py-name">ent</tt> <tt class="py-op">+=</tt> <tt class="py-op">-</tt> <tt class="py-number">1.0</tt> <tt class="py-op">*</tt> <tt class="py-name">a</tt> <tt class="py-op">*</tt> <tt class="py-name">digamma</tt><tt class="py-op">(</tt> <tt class="py-number">1.0</tt><tt class="py-op">+</tt><tt class="py-name">a</tt><tt class="py-op">)</tt> <tt class="py-comment"># FIXME: Check</tt> </tt>
<a name="L171"></a><tt class="py-lineno">171</tt>  <tt class="py-line">        <tt class="py-name">ent</tt> <tt class="py-op">/=</tt> <tt class="py-name">A</tt> </tt>
<a name="L172"></a><tt class="py-lineno">172</tt>  <tt class="py-line">        <tt class="py-name">ent</tt> <tt class="py-op">+=</tt> <tt class="py-name">digamma</tt><tt class="py-op">(</tt><tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">1.0</tt><tt class="py-op">)</tt> </tt>
<a name="L173"></a><tt class="py-lineno">173</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">ent</tt>     </tt>
</div><a name="L174"></a><tt class="py-lineno">174</tt>  <tt class="py-line"> </tt>
<a name="L175"></a><tt class="py-lineno">175</tt>  <tt class="py-line"> </tt>
<a name="L176"></a><tt class="py-lineno">176</tt>  <tt class="py-line"> </tt>
<a name="Dirichlet.variance_entropy"></a><div id="Dirichlet.variance_entropy-def"><a name="L177"></a><tt class="py-lineno">177</tt> <a class="py-toggle" href="#" id="Dirichlet.variance_entropy-toggle" onclick="return toggle('Dirichlet.variance_entropy');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#variance_entropy">variance_entropy</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.variance_entropy-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.variance_entropy-expanded"><a name="L178"></a><tt class="py-lineno">178</tt>  <tt class="py-line">        <tt class="py-docstring">"""Calculate the variance of the Dirichlet entropy. </tt> </tt>
<a name="L179"></a><tt class="py-lineno">179</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L180"></a><tt class="py-lineno">180</tt>  <tt class="py-line"><tt class="py-docstring">        Ref:</tt> </tt>
<a name="L181"></a><tt class="py-lineno">181</tt>  <tt class="py-line"><tt class="py-docstring">            Wolpert &amp; Wolf, PRE 53:6841-6854 (1996) Theorem 8</tt> </tt>
<a name="L182"></a><tt class="py-lineno">182</tt>  <tt class="py-line"><tt class="py-docstring">            (Warning: this paper contains typos.)</tt> </tt>
<a name="L183"></a><tt class="py-lineno">183</tt>  <tt class="py-line"><tt class="py-docstring">        """</tt> </tt>
<a name="L184"></a><tt class="py-lineno">184</tt>  <tt class="py-line">        <tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> </tt>
<a name="L185"></a><tt class="py-lineno">185</tt>  <tt class="py-line">        <tt class="py-name">A</tt> <tt class="py-op">=</tt> <tt class="py-name">float</tt><tt class="py-op">(</tt><tt class="py-name">sum</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> </tt>
<a name="L186"></a><tt class="py-lineno">186</tt>  <tt class="py-line">        <tt class="py-name">A2</tt> <tt class="py-op">=</tt> <tt class="py-name">A</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">1</tt><tt class="py-op">)</tt> </tt>
<a name="L187"></a><tt class="py-lineno">187</tt>  <tt class="py-line">        <tt class="py-name">L</tt> <tt class="py-op">=</tt> <tt class="py-name">len</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">)</tt> </tt>
<a name="L188"></a><tt class="py-lineno">188</tt>  <tt class="py-line">         </tt>
<a name="L189"></a><tt class="py-lineno">189</tt>  <tt class="py-line">        <tt class="py-name">dg1</tt> <tt class="py-op">=</tt> <tt class="py-name">zeros</tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L190"></a><tt class="py-lineno">190</tt>  <tt class="py-line">        <tt class="py-name">dg2</tt> <tt class="py-op">=</tt> <tt class="py-name">zeros</tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt> </tt>
<a name="L191"></a><tt class="py-lineno">191</tt>  <tt class="py-line">        <tt class="py-name">tg2</tt> <tt class="py-op">=</tt> <tt class="py-name">zeros</tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">,</tt> <tt class="py-name">float64</tt><tt class="py-op">)</tt>         </tt>
<a name="L192"></a><tt class="py-lineno">192</tt>  <tt class="py-line">         </tt>
<a name="L193"></a><tt class="py-lineno">193</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L194"></a><tt class="py-lineno">194</tt>  <tt class="py-line">            <tt class="py-name">dg1</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">digamma</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">1.0</tt><tt class="py-op">)</tt> </tt>
<a name="L195"></a><tt class="py-lineno">195</tt>  <tt class="py-line">            <tt class="py-name">dg2</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">digamma</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">2.0</tt><tt class="py-op">)</tt> </tt>
<a name="L196"></a><tt class="py-lineno">196</tt>  <tt class="py-line">            <tt class="py-name">tg2</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">=</tt> <tt class="py-name">trigamma</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">+</tt> <tt class="py-number">2.0</tt><tt class="py-op">)</tt> </tt>
<a name="L197"></a><tt class="py-lineno">197</tt>  <tt class="py-line"> </tt>
<a name="L198"></a><tt class="py-lineno">198</tt>  <tt class="py-line">        <tt class="py-name">dg_Ap2</tt> <tt class="py-op">=</tt> <tt class="py-name">digamma</tt><tt class="py-op">(</tt> <tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">2.</tt> <tt class="py-op">)</tt> </tt>
<a name="L199"></a><tt class="py-lineno">199</tt>  <tt class="py-line">        <tt class="py-name">tg_Ap2</tt> <tt class="py-op">=</tt> <tt class="py-name">trigamma</tt><tt class="py-op">(</tt> <tt class="py-name">A</tt><tt class="py-op">+</tt><tt class="py-number">2.</tt> <tt class="py-op">)</tt> </tt>
<a name="L200"></a><tt class="py-lineno">200</tt>  <tt class="py-line">         </tt>
<a name="L201"></a><tt class="py-lineno">201</tt>  <tt class="py-line">        <tt id="link-2" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-2', 'mean', 'link-0');">mean</a></tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-3" class="py-name" targets="Method weblogolib.logomath.Dirichlet.mean_entropy()=weblogolib.logomath.Dirichlet-class.html#mean_entropy"><a title="weblogolib.logomath.Dirichlet.mean_entropy" class="py-name" href="#" onclick="return doclink('link-3', 'mean_entropy', 'link-3');">mean_entropy</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
<a name="L202"></a><tt class="py-lineno">202</tt>  <tt class="py-line">        <tt class="py-name">var</tt> <tt class="py-op">=</tt> <tt class="py-number">0.0</tt> </tt>
<a name="L203"></a><tt class="py-lineno">203</tt>  <tt class="py-line">     </tt>
<a name="L204"></a><tt class="py-lineno">204</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L205"></a><tt class="py-lineno">205</tt>  <tt class="py-line">            <tt class="py-keyword">for</tt> <tt class="py-name">j</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">L</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L206"></a><tt class="py-lineno">206</tt>  <tt class="py-line">                <tt class="py-keyword">if</tt> <tt class="py-name">i</tt> <tt class="py-op">!=</tt> <tt class="py-name">j</tt> <tt class="py-op">:</tt> </tt>
<a name="L207"></a><tt class="py-lineno">207</tt>  <tt class="py-line">                    <tt class="py-name">var</tt> <tt class="py-op">+=</tt> <tt class="py-op">(</tt> </tt>
<a name="L208"></a><tt class="py-lineno">208</tt>  <tt class="py-line">                        <tt class="py-op">(</tt> <tt class="py-name">dg1</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">dg_Ap2</tt> <tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">dg1</tt><tt class="py-op">[</tt><tt class="py-name">j</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">dg_Ap2</tt> <tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">tg_Ap2</tt>  </tt>
<a name="L209"></a><tt class="py-lineno">209</tt>  <tt class="py-line">                        <tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">*</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">j</tt><tt class="py-op">]</tt> <tt class="py-op">)</tt> <tt class="py-op">/</tt> <tt class="py-name">A2</tt> </tt>
<a name="L210"></a><tt class="py-lineno">210</tt>  <tt class="py-line">                <tt class="py-keyword">else</tt> <tt class="py-op">:</tt>  </tt>
<a name="L211"></a><tt class="py-lineno">211</tt>  <tt class="py-line">                    <tt class="py-name">var</tt> <tt class="py-op">+=</tt> <tt class="py-op">(</tt> </tt>
<a name="L212"></a><tt class="py-lineno">212</tt>  <tt class="py-line">                        <tt class="py-op">(</tt> <tt class="py-name">dg2</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">dg_Ap2</tt> <tt class="py-op">)</tt> <tt class="py-op">**</tt><tt class="py-number">2</tt> <tt class="py-op">+</tt> <tt class="py-op">(</tt> <tt class="py-name">tg2</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">-</tt> <tt class="py-name">tg_Ap2</tt> <tt class="py-op">)</tt> </tt>
<a name="L213"></a><tt class="py-lineno">213</tt>  <tt class="py-line">                        <tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt> <tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt> <tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">[</tt><tt class="py-name">i</tt><tt class="py-op">]</tt><tt class="py-op">+</tt><tt class="py-number">1.</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> <tt class="py-op">/</tt> <tt class="py-name">A2</tt> </tt>
<a name="L214"></a><tt class="py-lineno">214</tt>  <tt class="py-line"> </tt>
<a name="L215"></a><tt class="py-lineno">215</tt>  <tt class="py-line">        <tt class="py-name">var</tt> <tt class="py-op">-=</tt> <tt id="link-4" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-4', 'mean', 'link-0');">mean</a></tt><tt class="py-op">**</tt><tt class="py-number">2</tt> </tt>
<a name="L216"></a><tt class="py-lineno">216</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">var</tt> </tt>
</div><a name="L217"></a><tt class="py-lineno">217</tt>  <tt class="py-line">         </tt>
<a name="L218"></a><tt class="py-lineno">218</tt>  <tt class="py-line">         </tt>
<a name="L219"></a><tt class="py-lineno">219</tt>  <tt class="py-line">         </tt>
<a name="Dirichlet.mean_relative_entropy"></a><div id="Dirichlet.mean_relative_entropy-def"><a name="L220"></a><tt class="py-lineno">220</tt> <a class="py-toggle" href="#" id="Dirichlet.mean_relative_entropy-toggle" onclick="return toggle('Dirichlet.mean_relative_entropy');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#mean_relative_entropy">mean_relative_entropy</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">pvec</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.mean_relative_entropy-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.mean_relative_entropy-expanded"><a name="L221"></a><tt class="py-lineno">221</tt>  <tt class="py-line">        <tt class="py-name">ln_p</tt> <tt class="py-op">=</tt> <tt class="py-name">na</tt><tt class="py-op">.</tt><tt class="py-name">log</tt><tt class="py-op">(</tt><tt class="py-name">pvec</tt><tt class="py-op">)</tt> </tt>
<a name="L222"></a><tt class="py-lineno">222</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-op">-</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-5" class="py-name" targets="Method weblogolib.logomath.Dirichlet.mean_x()=weblogolib.logomath.Dirichlet-class.html#mean_x"><a title="weblogolib.logomath.Dirichlet.mean_x" class="py-name" href="#" onclick="return doclink('link-5', 'mean_x', 'link-5');">mean_x</a></tt><tt class="py-op">(</tt><tt class="py-name">ln_p</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-6" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean_entropy" class="py-name" href="#" onclick="return doclink('link-6', 'mean_entropy', 'link-3');">mean_entropy</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt>  </tt>
</div><a name="L223"></a><tt class="py-lineno">223</tt>  <tt class="py-line">     </tt>
<a name="L224"></a><tt class="py-lineno">224</tt>  <tt class="py-line">     </tt>
<a name="Dirichlet.variance_relative_entropy"></a><div id="Dirichlet.variance_relative_entropy-def"><a name="L225"></a><tt class="py-lineno">225</tt> <a class="py-toggle" href="#" id="Dirichlet.variance_relative_entropy-toggle" onclick="return toggle('Dirichlet.variance_relative_entropy');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#variance_relative_entropy">variance_relative_entropy</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">pvec</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.variance_relative_entropy-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.variance_relative_entropy-expanded"><a name="L226"></a><tt class="py-lineno">226</tt>  <tt class="py-line">        <tt class="py-name">ln_p</tt> <tt class="py-op">=</tt> <tt class="py-name">na</tt><tt class="py-op">.</tt><tt class="py-name">log</tt><tt class="py-op">(</tt><tt class="py-name">pvec</tt><tt class="py-op">)</tt> </tt>
<a name="L227"></a><tt class="py-lineno">227</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-7" class="py-name" targets="Method weblogolib.logomath.Dirichlet.variance_x()=weblogolib.logomath.Dirichlet-class.html#variance_x"><a title="weblogolib.logomath.Dirichlet.variance_x" class="py-name" href="#" onclick="return doclink('link-7', 'variance_x', 'link-7');">variance_x</a></tt><tt class="py-op">(</tt><tt class="py-name">ln_p</tt><tt class="py-op">)</tt> <tt class="py-op">+</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-8" class="py-name" targets="Method weblogolib.logomath.Dirichlet.variance_entropy()=weblogolib.logomath.Dirichlet-class.html#variance_entropy"><a title="weblogolib.logomath.Dirichlet.variance_entropy" class="py-name" href="#" onclick="return doclink('link-8', 'variance_entropy', 'link-8');">variance_entropy</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt> </tt>
</div><a name="L228"></a><tt class="py-lineno">228</tt>  <tt class="py-line">     </tt>
<a name="L229"></a><tt class="py-lineno">229</tt>  <tt class="py-line">     </tt>
<a name="Dirichlet.interval_relative_entropy"></a><div id="Dirichlet.interval_relative_entropy-def"><a name="L230"></a><tt class="py-lineno">230</tt> <a class="py-toggle" href="#" id="Dirichlet.interval_relative_entropy-toggle" onclick="return toggle('Dirichlet.interval_relative_entropy');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Dirichlet-class.html#interval_relative_entropy">interval_relative_entropy</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">pvec</tt><tt class="py-op">,</tt> <tt class="py-param">frac</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Dirichlet.interval_relative_entropy-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Dirichlet.interval_relative_entropy-expanded"><a name="L231"></a><tt class="py-lineno">231</tt>  <tt class="py-line">        <tt id="link-9" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-9', 'mean', 'link-0');">mean</a></tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-10" class="py-name" targets="Method weblogolib.logomath.Dirichlet.mean_relative_entropy()=weblogolib.logomath.Dirichlet-class.html#mean_relative_entropy"><a title="weblogolib.logomath.Dirichlet.mean_relative_entropy" class="py-name" href="#" onclick="return doclink('link-10', 'mean_relative_entropy', 'link-10');">mean_relative_entropy</a></tt><tt class="py-op">(</tt><tt class="py-name">pvec</tt><tt class="py-op">)</tt>  </tt>
<a name="L232"></a><tt class="py-lineno">232</tt>  <tt class="py-line">        <tt id="link-11" class="py-name" targets="Method weblogolib.logomath.Gamma.variance()=weblogolib.logomath.Gamma-class.html#variance"><a title="weblogolib.logomath.Gamma.variance" class="py-name" href="#" onclick="return doclink('link-11', 'variance', 'link-11');">variance</a></tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-12" class="py-name" targets="Method weblogolib.logomath.Dirichlet.variance_relative_entropy()=weblogolib.logomath.Dirichlet-class.html#variance_relative_entropy"><a title="weblogolib.logomath.Dirichlet.variance_relative_entropy" class="py-name" href="#" onclick="return doclink('link-12', 'variance_relative_entropy', 'link-12');">variance_relative_entropy</a></tt><tt class="py-op">(</tt><tt class="py-name">pvec</tt><tt class="py-op">)</tt>  </tt>
<a name="L233"></a><tt class="py-lineno">233</tt>  <tt class="py-line">        <tt class="py-name">sd</tt> <tt class="py-op">=</tt> <tt class="py-name">sqrt</tt><tt class="py-op">(</tt><tt id="link-13" class="py-name"><a title="weblogolib.logomath.Gamma.variance" class="py-name" href="#" onclick="return doclink('link-13', 'variance', 'link-11');">variance</a></tt><tt class="py-op">)</tt> </tt>
<a name="L234"></a><tt class="py-lineno">234</tt>  <tt class="py-line">         </tt>
<a name="L235"></a><tt class="py-lineno">235</tt>  <tt class="py-line">        <tt class="py-comment"># If the variance is small, use the standard 95% </tt> </tt>
<a name="L236"></a><tt class="py-lineno">236</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-comment"># confidence interval: mean +/- 1.96 * sd</tt> </tt>
<a name="L237"></a><tt class="py-lineno">237</tt>  <tt class="py-line"><tt class="py-comment"></tt>        <tt class="py-keyword">if</tt> <tt id="link-14" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-14', 'mean', 'link-0');">mean</a></tt><tt class="py-op">/</tt><tt class="py-name">sd</tt> <tt class="py-op">&gt;</tt><tt class="py-number">3.0</tt> <tt class="py-op">:</tt> </tt>
<a name="L238"></a><tt class="py-lineno">238</tt>  <tt class="py-line">            <tt class="py-keyword">return</tt> <tt class="py-name">max</tt><tt class="py-op">(</tt><tt class="py-number">0.0</tt><tt class="py-op">,</tt> <tt id="link-15" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-15', 'mean', 'link-0');">mean</a></tt> <tt class="py-op">-</tt> <tt class="py-name">sd</tt><tt class="py-op">*</tt><tt class="py-number">1.96</tt><tt class="py-op">)</tt><tt class="py-op">,</tt> <tt id="link-16" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-16', 'mean', 'link-0');">mean</a></tt> <tt class="py-op">+</tt> <tt class="py-name">sd</tt><tt class="py-op">*</tt><tt class="py-number">1.96</tt> </tt>
<a name="L239"></a><tt class="py-lineno">239</tt>  <tt class="py-line">         </tt>
<a name="L240"></a><tt class="py-lineno">240</tt>  <tt class="py-line">        <tt class="py-name">g</tt> <tt class="py-op">=</tt> <tt id="link-17" class="py-name" targets="Class weblogolib.logomath.Gamma=weblogolib.logomath.Gamma-class.html"><a title="weblogolib.logomath.Gamma" class="py-name" href="#" onclick="return doclink('link-17', 'Gamma', 'link-17');">Gamma</a></tt><tt class="py-op">.</tt><tt id="link-18" class="py-name" targets="Class Method weblogolib.logomath.Gamma.from_mean_variance()=weblogolib.logomath.Gamma-class.html#from_mean_variance"><a title="weblogolib.logomath.Gamma.from_mean_variance" class="py-name" href="#" onclick="return doclink('link-18', 'from_mean_variance', 'link-18');">from_mean_variance</a></tt><tt class="py-op">(</tt><tt id="link-19" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-19', 'mean', 'link-0');">mean</a></tt><tt class="py-op">,</tt> <tt id="link-20" class="py-name"><a title="weblogolib.logomath.Gamma.variance" class="py-name" href="#" onclick="return doclink('link-20', 'variance', 'link-11');">variance</a></tt><tt class="py-op">)</tt> </tt>
<a name="L241"></a><tt class="py-lineno">241</tt>  <tt class="py-line">        <tt class="py-name">low_limit</tt> <tt class="py-op">=</tt> <tt class="py-name">g</tt><tt class="py-op">.</tt><tt id="link-21" class="py-name" targets="Method weblogolib.logomath.Gamma.inverse_cdf()=weblogolib.logomath.Gamma-class.html#inverse_cdf"><a title="weblogolib.logomath.Gamma.inverse_cdf" class="py-name" href="#" onclick="return doclink('link-21', 'inverse_cdf', 'link-21');">inverse_cdf</a></tt><tt class="py-op">(</tt> <tt class="py-op">(</tt><tt class="py-number">1.</tt><tt class="py-op">-</tt><tt class="py-name">frac</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-number">2.</tt><tt class="py-op">)</tt> </tt>
<a name="L242"></a><tt class="py-lineno">242</tt>  <tt class="py-line">        <tt class="py-name">high_limit</tt> <tt class="py-op">=</tt> <tt class="py-name">g</tt><tt class="py-op">.</tt><tt id="link-22" class="py-name"><a title="weblogolib.logomath.Gamma.inverse_cdf" class="py-name" href="#" onclick="return doclink('link-22', 'inverse_cdf', 'link-21');">inverse_cdf</a></tt><tt class="py-op">(</tt> <tt class="py-number">1.</tt> <tt class="py-op">-</tt> <tt class="py-op">(</tt><tt class="py-number">1.</tt><tt class="py-op">-</tt><tt class="py-name">frac</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-number">2.</tt> <tt class="py-op">)</tt> </tt>
<a name="L243"></a><tt class="py-lineno">243</tt>  <tt class="py-line">         </tt>
<a name="L244"></a><tt class="py-lineno">244</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">low_limit</tt><tt class="py-op">,</tt> <tt class="py-name">high_limit</tt> </tt>
</div></div><a name="L245"></a><tt class="py-lineno">245</tt>  <tt class="py-line"> </tt>
<a name="L246"></a><tt class="py-lineno">246</tt>  <tt class="py-line"> </tt>
<a name="Gamma"></a><div id="Gamma-def"><a name="L247"></a><tt class="py-lineno">247</tt> <a class="py-toggle" href="#" id="Gamma-toggle" onclick="return toggle('Gamma');">-</a><tt class="py-line"><tt class="py-keyword">class</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html">Gamma</a><tt class="py-op">(</tt><tt class="py-base-class">object</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="Gamma-expanded"><a name="L248"></a><tt class="py-lineno">248</tt>  <tt class="py-line">    <tt class="py-docstring">"""The gamma probability distribution. (Not to be confused with the</tt> </tt>
<a name="L249"></a><tt class="py-lineno">249</tt>  <tt class="py-line"><tt class="py-docstring">    gamma function.)</tt> </tt>
<a name="L250"></a><tt class="py-lineno">250</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L251"></a><tt class="py-lineno">251</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L252"></a><tt class="py-lineno">252</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L253"></a><tt class="py-lineno">253</tt>  <tt class="py-line">    <tt class="py-name">__slots__</tt> <tt class="py-op">=</tt> <tt class="py-string">'alpha'</tt><tt class="py-op">,</tt> <tt class="py-string">'beta'</tt> </tt>
<a name="L254"></a><tt class="py-lineno">254</tt>  <tt class="py-line">     </tt>
<a name="Gamma.__init__"></a><div id="Gamma.__init__-def"><a name="L255"></a><tt class="py-lineno">255</tt> <a class="py-toggle" href="#" id="Gamma.__init__-toggle" onclick="return toggle('Gamma.__init__');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#__init__">__init__</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">alpha</tt><tt class="py-op">,</tt> <tt class="py-param">beta</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.__init__-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.__init__-expanded"><a name="L256"></a><tt class="py-lineno">256</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">alpha</tt> <tt class="py-op">&lt;=</tt><tt class="py-number">0.0</tt> <tt class="py-op">:</tt> </tt>
<a name="L257"></a><tt class="py-lineno">257</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">"alpha must be positive"</tt><tt class="py-op">)</tt> </tt>
<a name="L258"></a><tt class="py-lineno">258</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">beta</tt> <tt class="py-op">&lt;=</tt><tt class="py-number">0.0</tt> <tt class="py-op">:</tt> </tt>
<a name="L259"></a><tt class="py-lineno">259</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ValueError</tt><tt class="py-op">(</tt><tt class="py-string">"beta must be positive"</tt><tt class="py-op">)</tt> </tt>
<a name="L260"></a><tt class="py-lineno">260</tt>  <tt class="py-line">        <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt class="py-name">alpha</tt> </tt>
<a name="L261"></a><tt class="py-lineno">261</tt>  <tt class="py-line">        <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt> <tt class="py-op">=</tt> <tt class="py-name">beta</tt> </tt>
</div><a name="L262"></a><tt class="py-lineno">262</tt>  <tt class="py-line">         </tt>
<a name="L263"></a><tt class="py-lineno">263</tt>  <tt class="py-line"> </tt>
<a name="Gamma.from_shape_scale"></a><div id="Gamma.from_shape_scale-def"><a name="L264"></a><tt class="py-lineno">264</tt> <a class="py-toggle" href="#" id="Gamma.from_shape_scale-toggle" onclick="return toggle('Gamma.from_shape_scale');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#from_shape_scale">from_shape_scale</a><tt class="py-op">(</tt><tt class="py-param">cls</tt><tt class="py-op">,</tt> <tt class="py-param">shape</tt><tt class="py-op">,</tt> <tt class="py-param">scale</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.from_shape_scale-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.from_shape_scale-expanded"><a name="L265"></a><tt class="py-lineno">265</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">cls</tt><tt class="py-op">(</tt> <tt class="py-name">shape</tt><tt class="py-op">,</tt> <tt class="py-number">1.</tt><tt class="py-op">/</tt><tt class="py-name">scale</tt><tt class="py-op">)</tt> </tt>
</div><a name="L266"></a><tt class="py-lineno">266</tt>  <tt class="py-line">    <tt id="link-23" class="py-name" targets="Class Method weblogolib.logomath.Gamma.from_shape_scale()=weblogolib.logomath.Gamma-class.html#from_shape_scale"><a title="weblogolib.logomath.Gamma.from_shape_scale" class="py-name" href="#" onclick="return doclink('link-23', 'from_shape_scale', 'link-23');">from_shape_scale</a></tt> <tt class="py-op">=</tt> <tt class="py-name">classmethod</tt><tt class="py-op">(</tt><tt id="link-24" class="py-name"><a title="weblogolib.logomath.Gamma.from_shape_scale" class="py-name" href="#" onclick="return doclink('link-24', 'from_shape_scale', 'link-23');">from_shape_scale</a></tt><tt class="py-op">)</tt> </tt>
<a name="L267"></a><tt class="py-lineno">267</tt>  <tt class="py-line">     </tt>
<a name="L268"></a><tt class="py-lineno">268</tt>  <tt class="py-line"> </tt>
<a name="Gamma.from_mean_variance"></a><div id="Gamma.from_mean_variance-def"><a name="L269"></a><tt class="py-lineno">269</tt> <a class="py-toggle" href="#" id="Gamma.from_mean_variance-toggle" onclick="return toggle('Gamma.from_mean_variance');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#from_mean_variance">from_mean_variance</a><tt class="py-op">(</tt><tt class="py-param">cls</tt><tt class="py-op">,</tt> <tt class="py-param">mean</tt><tt class="py-op">,</tt> <tt class="py-param">variance</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.from_mean_variance-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.from_mean_variance-expanded"><a name="L270"></a><tt class="py-lineno">270</tt>  <tt class="py-line">        <tt class="py-name">alpha</tt> <tt class="py-op">=</tt> <tt id="link-25" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-25', 'mean', 'link-0');">mean</a></tt> <tt class="py-op">**</tt><tt class="py-number">2</tt> <tt class="py-op">/</tt> <tt id="link-26" class="py-name"><a title="weblogolib.logomath.Gamma.variance" class="py-name" href="#" onclick="return doclink('link-26', 'variance', 'link-11');">variance</a></tt> </tt>
<a name="L271"></a><tt class="py-lineno">271</tt>  <tt class="py-line">        <tt class="py-name">beta</tt> <tt class="py-op">=</tt> <tt class="py-name">alpha</tt><tt class="py-op">/</tt><tt id="link-27" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-27', 'mean', 'link-0');">mean</a></tt> </tt>
<a name="L272"></a><tt class="py-lineno">272</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">cls</tt><tt class="py-op">(</tt><tt class="py-name">alpha</tt><tt class="py-op">,</tt> <tt class="py-name">beta</tt><tt class="py-op">)</tt> </tt>
</div><a name="L273"></a><tt class="py-lineno">273</tt>  <tt class="py-line">    <tt id="link-28" class="py-name"><a title="weblogolib.logomath.Gamma.from_mean_variance" class="py-name" href="#" onclick="return doclink('link-28', 'from_mean_variance', 'link-18');">from_mean_variance</a></tt> <tt class="py-op">=</tt> <tt class="py-name">classmethod</tt><tt class="py-op">(</tt><tt id="link-29" class="py-name"><a title="weblogolib.logomath.Gamma.from_mean_variance" class="py-name" href="#" onclick="return doclink('link-29', 'from_mean_variance', 'link-18');">from_mean_variance</a></tt><tt class="py-op">)</tt> </tt>
<a name="L274"></a><tt class="py-lineno">274</tt>  <tt class="py-line"> </tt>
<a name="Gamma.mean"></a><div id="Gamma.mean-def"><a name="L275"></a><tt class="py-lineno">275</tt> <a class="py-toggle" href="#" id="Gamma.mean-toggle" onclick="return toggle('Gamma.mean');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#mean">mean</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.mean-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.mean-expanded"><a name="L276"></a><tt class="py-lineno">276</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> <tt class="py-op">/</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt> </tt>
</div><a name="L277"></a><tt class="py-lineno">277</tt>  <tt class="py-line">         </tt>
<a name="Gamma.variance"></a><div id="Gamma.variance-def"><a name="L278"></a><tt class="py-lineno">278</tt> <a class="py-toggle" href="#" id="Gamma.variance-toggle" onclick="return toggle('Gamma.variance');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#variance">variance</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.variance-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.variance-expanded"><a name="L279"></a><tt class="py-lineno">279</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> <tt class="py-op">/</tt> <tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt><tt class="py-op">**</tt><tt class="py-number">2</tt><tt class="py-op">)</tt> </tt>
</div><a name="L280"></a><tt class="py-lineno">280</tt>  <tt class="py-line"> </tt>
<a name="L281"></a><tt class="py-lineno">281</tt>  <tt class="py-line">         </tt>
<a name="Gamma.sample"></a><div id="Gamma.sample-def"><a name="L282"></a><tt class="py-lineno">282</tt> <a class="py-toggle" href="#" id="Gamma.sample-toggle" onclick="return toggle('Gamma.sample');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#sample">sample</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.sample-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.sample-expanded"><a name="L283"></a><tt class="py-lineno">283</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">random</tt><tt class="py-op">.</tt><tt class="py-name">gammavariate</tt><tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt><tt class="py-op">,</tt> <tt class="py-number">1.</tt><tt class="py-op">/</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt><tt class="py-op">)</tt>  </tt>
</div><a name="L284"></a><tt class="py-lineno">284</tt>  <tt class="py-line"> </tt>
<a name="Gamma.pdf"></a><div id="Gamma.pdf-def"><a name="L285"></a><tt class="py-lineno">285</tt> <a class="py-toggle" href="#" id="Gamma.pdf-toggle" onclick="return toggle('Gamma.pdf');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#pdf">pdf</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.pdf-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.pdf-expanded"><a name="L286"></a><tt class="py-lineno">286</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">x</tt><tt class="py-op">==</tt><tt class="py-number">0.0</tt> <tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-number">0.0</tt> </tt>
<a name="L287"></a><tt class="py-lineno">287</tt>  <tt class="py-line">        <tt class="py-name">a</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt> </tt>
<a name="L288"></a><tt class="py-lineno">288</tt>  <tt class="py-line">        <tt class="py-name">b</tt> <tt class="py-op">=</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt> </tt>
<a name="L289"></a><tt class="py-lineno">289</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">**</tt><tt class="py-op">(</tt><tt class="py-name">a</tt><tt class="py-op">-</tt><tt class="py-number">1.</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">*</tt> <tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-op">-</tt> <tt class="py-name">b</tt><tt class="py-op">*</tt><tt class="py-name">x</tt> <tt class="py-op">)</tt><tt class="py-op">*</tt> <tt class="py-op">(</tt><tt class="py-name">b</tt><tt class="py-op">**</tt><tt class="py-name">a</tt><tt class="py-op">)</tt> <tt class="py-op">/</tt> <tt class="py-name">gamma</tt><tt class="py-op">(</tt><tt class="py-name">a</tt><tt class="py-op">)</tt>  </tt>
</div><a name="L290"></a><tt class="py-lineno">290</tt>  <tt class="py-line">         </tt>
<a name="Gamma.cdf"></a><div id="Gamma.cdf-def"><a name="L291"></a><tt class="py-lineno">291</tt> <a class="py-toggle" href="#" id="Gamma.cdf-toggle" onclick="return toggle('Gamma.cdf');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#cdf">cdf</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.cdf-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.cdf-expanded"><a name="L292"></a><tt class="py-lineno">292</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-number">1.0</tt><tt class="py-op">-</tt><tt class="py-name">normalized_incomplete_gamma</tt><tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">alpha</tt><tt class="py-op">,</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt class="py-name">beta</tt><tt class="py-op">*</tt><tt class="py-name">x</tt><tt class="py-op">)</tt> </tt>
</div><a name="L293"></a><tt class="py-lineno">293</tt>  <tt class="py-line"> </tt>
<a name="Gamma.inverse_cdf"></a><div id="Gamma.inverse_cdf-def"><a name="L294"></a><tt class="py-lineno">294</tt> <a class="py-toggle" href="#" id="Gamma.inverse_cdf-toggle" onclick="return toggle('Gamma.inverse_cdf');">-</a><tt class="py-line">    <tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath.Gamma-class.html#inverse_cdf">inverse_cdf</a><tt class="py-op">(</tt><tt class="py-param">self</tt><tt class="py-op">,</tt> <tt class="py-param">p</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
</div><div id="Gamma.inverse_cdf-collapsed" style="display:none;" pad="+++" indent="++++++++"></div><div id="Gamma.inverse_cdf-expanded"><a name="L295"></a><tt class="py-lineno">295</tt>  <tt class="py-line">        <tt class="py-keyword">def</tt> <tt class="py-def-name">rootof</tt><tt class="py-op">(</tt><tt class="py-param">x</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L296"></a><tt class="py-lineno">296</tt>  <tt class="py-line">            <tt class="py-keyword">return</tt> <tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-30" class="py-name" targets="Method weblogolib.logomath.Gamma.cdf()=weblogolib.logomath.Gamma-class.html#cdf"><a title="weblogolib.logomath.Gamma.cdf" class="py-name" href="#" onclick="return doclink('link-30', 'cdf', 'link-30');">cdf</a></tt><tt class="py-op">(</tt><tt class="py-name">exp</tt><tt class="py-op">(</tt><tt class="py-name">x</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">-</tt> <tt class="py-name">p</tt> </tt>
</div><a name="L297"></a><tt class="py-lineno">297</tt>  <tt class="py-line">     </tt>
<a name="L298"></a><tt class="py-lineno">298</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">exp</tt><tt class="py-op">(</tt><tt id="link-31" class="py-name" targets="Function weblogolib.logomath.find_root()=weblogolib.logomath-module.html#find_root"><a title="weblogolib.logomath.find_root" class="py-name" href="#" onclick="return doclink('link-31', 'find_root', 'link-31');">find_root</a></tt><tt class="py-op">(</tt>  <tt class="py-name">rootof</tt><tt class="py-op">,</tt> <tt class="py-name">log</tt><tt class="py-op">(</tt><tt class="py-name">self</tt><tt class="py-op">.</tt><tt id="link-32" class="py-name"><a title="weblogolib.logomath.Dirichlet.mean
weblogolib.logomath.Gamma.mean" class="py-name" href="#" onclick="return doclink('link-32', 'mean', 'link-0');">mean</a></tt><tt class="py-op">(</tt><tt class="py-op">)</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
</div></div><a name="L299"></a><tt class="py-lineno">299</tt>  <tt class="py-line"> </tt>
<a name="L300"></a><tt class="py-lineno">300</tt>  <tt class="py-line"><tt class="py-comment">#</tt> </tt>
<a name="L301"></a><tt class="py-lineno">301</tt>  <tt class="py-line"><tt class="py-comment"></tt> </tt>
<a name="find_root"></a><div id="find_root-def"><a name="L302"></a><tt class="py-lineno">302</tt> <a class="py-toggle" href="#" id="find_root-toggle" onclick="return toggle('find_root');">-</a><tt class="py-line"><tt class="py-keyword">def</tt> <a class="py-def-name" href="weblogolib.logomath-module.html#find_root">find_root</a><tt class="py-op">(</tt><tt class="py-param">f</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">,</tt> <tt class="py-param">y</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-param">fprime</tt><tt class="py-op">=</tt><tt class="py-name">None</tt><tt class="py-op">,</tt> <tt class="py-param">tolerance</tt><tt class="py-op">=</tt><tt class="py-number">1.48e-8</tt><tt class="py-op">,</tt> <tt class="py-param">max_iterations</tt><tt class="py-op">=</tt><tt class="py-number">50</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
</div><div id="find_root-collapsed" style="display:none;" pad="+++" indent="++++"></div><div id="find_root-expanded"><a name="L303"></a><tt class="py-lineno">303</tt>  <tt class="py-line">    <tt class="py-docstring">"""Return argument 'x' of the function f(x), such that f(x)=0 to</tt> </tt>
<a name="L304"></a><tt class="py-lineno">304</tt>  <tt class="py-line"><tt class="py-docstring">    within the given tolerance. </tt> </tt>
<a name="L305"></a><tt class="py-lineno">305</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L306"></a><tt class="py-lineno">306</tt>  <tt class="py-line"><tt class="py-docstring">    f : The function to optimize, f(x)</tt> </tt>
<a name="L307"></a><tt class="py-lineno">307</tt>  <tt class="py-line"><tt class="py-docstring">    x : The initial guess</tt> </tt>
<a name="L308"></a><tt class="py-lineno">308</tt>  <tt class="py-line"><tt class="py-docstring">    y : An optional second guess that shoudl bracket the root.</tt> </tt>
<a name="L309"></a><tt class="py-lineno">309</tt>  <tt class="py-line"><tt class="py-docstring">    fprime : The derivate of f'(x) (Optional)</tt> </tt>
<a name="L310"></a><tt class="py-lineno">310</tt>  <tt class="py-line"><tt class="py-docstring">    tolerance : The error bounds </tt> </tt>
<a name="L311"></a><tt class="py-lineno">311</tt>  <tt class="py-line"><tt class="py-docstring">    max_iterations : Maximum number of iterations</tt> </tt>
<a name="L312"></a><tt class="py-lineno">312</tt>  <tt class="py-line"><tt class="py-docstring">    </tt> </tt>
<a name="L313"></a><tt class="py-lineno">313</tt>  <tt class="py-line"><tt class="py-docstring">    Raises:</tt> </tt>
<a name="L314"></a><tt class="py-lineno">314</tt>  <tt class="py-line"><tt class="py-docstring">        ArithmeticError :</tt> </tt>
<a name="L315"></a><tt class="py-lineno">315</tt>  <tt class="py-line"><tt class="py-docstring">            Failure to converge to the given tolerence</tt> </tt>
<a name="L316"></a><tt class="py-lineno">316</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L317"></a><tt class="py-lineno">317</tt>  <tt class="py-line"><tt class="py-docstring">    Notes:</tt> </tt>
<a name="L318"></a><tt class="py-lineno">318</tt>  <tt class="py-line"><tt class="py-docstring">        Uses Newton-Raphson algorihtm if f'(x) is given, else uses bisect if</tt> </tt>
<a name="L319"></a><tt class="py-lineno">319</tt>  <tt class="py-line"><tt class="py-docstring">        y is given and brackets the root, else uses secant. </tt> </tt>
<a name="L320"></a><tt class="py-lineno">320</tt>  <tt class="py-line"><tt class="py-docstring"></tt> </tt>
<a name="L321"></a><tt class="py-lineno">321</tt>  <tt class="py-line"><tt class="py-docstring">    Status : Beta (Not fully tested)</tt> </tt>
<a name="L322"></a><tt class="py-lineno">322</tt>  <tt class="py-line"><tt class="py-docstring">    """</tt> </tt>
<a name="L323"></a><tt class="py-lineno">323</tt>  <tt class="py-line"> </tt>
<a name="L324"></a><tt class="py-lineno">324</tt>  <tt class="py-line">   </tt>
<a name="L325"></a><tt class="py-lineno">325</tt>  <tt class="py-line">    <tt class="py-keyword">def</tt> <tt class="py-def-name">secant</tt> <tt class="py-op">(</tt><tt class="py-param">f</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">,</tt> <tt class="py-param">tolerance</tt><tt class="py-op">,</tt> <tt class="py-param">max_iterations</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L326"></a><tt class="py-lineno">326</tt>  <tt class="py-line">        <tt class="py-name">x0</tt> <tt class="py-op">=</tt> <tt class="py-name">x</tt> </tt>
<a name="L327"></a><tt class="py-lineno">327</tt>  <tt class="py-line">        <tt class="py-name">x1</tt> <tt class="py-op">=</tt> <tt class="py-name">x0</tt><tt class="py-op">+</tt> <tt class="py-number">1e-4</tt> </tt>
<a name="L328"></a><tt class="py-lineno">328</tt>  <tt class="py-line">        <tt class="py-name">v0</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">x0</tt><tt class="py-op">)</tt> </tt>
<a name="L329"></a><tt class="py-lineno">329</tt>  <tt class="py-line">        <tt class="py-name">v1</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt> </tt>
<a name="L330"></a><tt class="py-lineno">330</tt>  <tt class="py-line">        <tt class="py-name">x2</tt> <tt class="py-op">=</tt> <tt class="py-number">0</tt> </tt>
<a name="L331"></a><tt class="py-lineno">331</tt>  <tt class="py-line">         </tt>
<a name="L332"></a><tt class="py-lineno">332</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L333"></a><tt class="py-lineno">333</tt>  <tt class="py-line">            <tt class="py-comment"># print x0, x1, v0, v1, x2-x0</tt> </tt>
<a name="L334"></a><tt class="py-lineno">334</tt>  <tt class="py-line"><tt class="py-comment"></tt>            <tt class="py-name">x2</tt> <tt class="py-op">=</tt> <tt class="py-name">x1</tt> <tt class="py-op">-</tt> <tt class="py-name">v1</tt><tt class="py-op">*</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">-</tt><tt class="py-name">x0</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-op">(</tt><tt class="py-name">v1</tt><tt class="py-op">-</tt><tt class="py-name">v0</tt><tt class="py-op">)</tt> </tt>
<a name="L335"></a><tt class="py-lineno">335</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">abs</tt><tt class="py-op">(</tt><tt class="py-name">x2</tt><tt class="py-op">-</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt> <tt class="py-op">&lt;</tt> <tt class="py-name">tolerance</tt> <tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-name">x2</tt>         </tt>
<a name="L336"></a><tt class="py-lineno">336</tt>  <tt class="py-line">            <tt class="py-name">x0</tt> <tt class="py-op">=</tt> <tt class="py-name">x1</tt> </tt>
<a name="L337"></a><tt class="py-lineno">337</tt>  <tt class="py-line">            <tt class="py-name">v0</tt> <tt class="py-op">=</tt> <tt class="py-name">v1</tt>         </tt>
<a name="L338"></a><tt class="py-lineno">338</tt>  <tt class="py-line">            <tt class="py-name">x1</tt> <tt class="py-op">=</tt> <tt class="py-name">x2</tt> </tt>
<a name="L339"></a><tt class="py-lineno">339</tt>  <tt class="py-line">            <tt class="py-name">v1</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt> </tt>
<a name="L340"></a><tt class="py-lineno">340</tt>  <tt class="py-line"> </tt>
<a name="L341"></a><tt class="py-lineno">341</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ArithmeticError</tt><tt class="py-op">(</tt> </tt>
<a name="L342"></a><tt class="py-lineno">342</tt>  <tt class="py-line">            <tt class="py-string">"Failed to converge after %d iterations, value is %f"</tt> \ </tt>
<a name="L343"></a><tt class="py-lineno">343</tt>  <tt class="py-line">                <tt class="py-op">%</tt> <tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">,</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
</div><a name="L344"></a><tt class="py-lineno">344</tt>  <tt class="py-line"> </tt>
<a name="L345"></a><tt class="py-lineno">345</tt>  <tt class="py-line"> </tt>
<a name="L346"></a><tt class="py-lineno">346</tt>  <tt class="py-line">    <tt class="py-keyword">def</tt> <tt class="py-def-name">bisect</tt><tt class="py-op">(</tt><tt class="py-param">f</tt><tt class="py-op">,</tt> <tt class="py-param">a</tt><tt class="py-op">,</tt> <tt class="py-param">b</tt><tt class="py-op">,</tt> <tt class="py-param">tolerance</tt><tt class="py-op">,</tt> <tt class="py-param">max_iterations</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L347"></a><tt class="py-lineno">347</tt>  <tt class="py-line">        <tt class="py-name">fa</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">a</tt><tt class="py-op">)</tt> </tt>
<a name="L348"></a><tt class="py-lineno">348</tt>  <tt class="py-line">        <tt class="py-name">fb</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">b</tt><tt class="py-op">)</tt> </tt>
<a name="L349"></a><tt class="py-lineno">349</tt>  <tt class="py-line"> </tt>
<a name="L350"></a><tt class="py-lineno">350</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">fa</tt><tt class="py-op">==</tt><tt class="py-number">0</tt><tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-name">a</tt> </tt>
<a name="L351"></a><tt class="py-lineno">351</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">fb</tt><tt class="py-op">==</tt><tt class="py-number">0</tt><tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-name">b</tt> </tt>
<a name="L352"></a><tt class="py-lineno">352</tt>  <tt class="py-line">                 </tt>
<a name="L353"></a><tt class="py-lineno">353</tt>  <tt class="py-line">        <tt class="py-keyword">if</tt> <tt class="py-name">fa</tt><tt class="py-op">*</tt><tt class="py-name">fb</tt> <tt class="py-op">&gt;</tt><tt class="py-number">0</tt> <tt class="py-op">:</tt> </tt>
<a name="L354"></a><tt class="py-lineno">354</tt>  <tt class="py-line">            <tt class="py-keyword">raise</tt> <tt class="py-name">ArithmeticError</tt><tt class="py-op">(</tt><tt class="py-string">"Start points do not bracket root."</tt><tt class="py-op">)</tt> </tt>
<a name="L355"></a><tt class="py-lineno">355</tt>  <tt class="py-line">         </tt>
<a name="L356"></a><tt class="py-lineno">356</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">)</tt><tt class="py-op">:</tt> </tt>
<a name="L357"></a><tt class="py-lineno">357</tt>  <tt class="py-line">            <tt class="py-comment">#print a,b, fa, fb</tt> </tt>
<a name="L358"></a><tt class="py-lineno">358</tt>  <tt class="py-line"><tt class="py-comment"></tt>            <tt class="py-name">delta</tt> <tt class="py-op">=</tt> <tt class="py-name">b</tt><tt class="py-op">-</tt><tt class="py-name">a</tt> </tt>
<a name="L359"></a><tt class="py-lineno">359</tt>  <tt class="py-line">            <tt class="py-name">xm</tt> <tt class="py-op">=</tt> <tt class="py-name">a</tt> <tt class="py-op">+</tt> <tt class="py-number">0.5</tt><tt class="py-op">*</tt><tt class="py-name">delta</tt> <tt class="py-comment"># Minimize roundoff in computing the midpoint</tt> </tt>
<a name="L360"></a><tt class="py-lineno">360</tt>  <tt class="py-line">            <tt class="py-name">fm</tt> <tt class="py-op">=</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">xm</tt><tt class="py-op">)</tt> </tt>
<a name="L361"></a><tt class="py-lineno">361</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">delta</tt> <tt class="py-op">&lt;</tt> <tt class="py-name">tolerance</tt> <tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-name">xm</tt> </tt>
<a name="L362"></a><tt class="py-lineno">362</tt>  <tt class="py-line">             </tt>
<a name="L363"></a><tt class="py-lineno">363</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">fm</tt> <tt class="py-op">*</tt> <tt class="py-name">fa</tt> <tt class="py-op">&gt;</tt><tt class="py-number">0</tt> <tt class="py-op">:</tt> <tt class="py-comment">#   Root lies in interval [xm,b], replace a</tt> </tt>
<a name="L364"></a><tt class="py-lineno">364</tt>  <tt class="py-line">                <tt class="py-name">a</tt> <tt class="py-op">=</tt> <tt class="py-name">xm</tt> </tt>
<a name="L365"></a><tt class="py-lineno">365</tt>  <tt class="py-line">                <tt class="py-name">fa</tt> <tt class="py-op">=</tt> <tt class="py-name">fm</tt> </tt>
<a name="L366"></a><tt class="py-lineno">366</tt>  <tt class="py-line">            <tt class="py-keyword">else</tt> <tt class="py-op">:</tt> <tt class="py-comment">#   Root lies in interval [a,xm], replace b</tt> </tt>
<a name="L367"></a><tt class="py-lineno">367</tt>  <tt class="py-line">                <tt class="py-name">b</tt> <tt class="py-op">=</tt> <tt class="py-name">xm</tt> </tt>
<a name="L368"></a><tt class="py-lineno">368</tt>  <tt class="py-line">                <tt class="py-name">fb</tt> <tt class="py-op">=</tt> <tt class="py-name">fm</tt> </tt>
<a name="L369"></a><tt class="py-lineno">369</tt>  <tt class="py-line"> </tt>
<a name="L370"></a><tt class="py-lineno">370</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ArithmeticError</tt><tt class="py-op">(</tt> </tt>
<a name="L371"></a><tt class="py-lineno">371</tt>  <tt class="py-line">            <tt class="py-string">"Failed to converge after %d iterations, value is %f"</tt> \ </tt>
<a name="L372"></a><tt class="py-lineno">372</tt>  <tt class="py-line">                <tt class="py-op">%</tt> <tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">,</tt> <tt class="py-name">x</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
</div><a name="L373"></a><tt class="py-lineno">373</tt>  <tt class="py-line">     </tt>
<a name="L374"></a><tt class="py-lineno">374</tt>  <tt class="py-line">     </tt>
<a name="L375"></a><tt class="py-lineno">375</tt>  <tt class="py-line">    <tt class="py-keyword">def</tt> <tt class="py-def-name">newton</tt><tt class="py-op">(</tt><tt class="py-param">f</tt><tt class="py-op">,</tt> <tt class="py-param">x</tt><tt class="py-op">,</tt> <tt class="py-param">fprime</tt><tt class="py-op">,</tt> <tt class="py-param">tolerance</tt><tt class="py-op">,</tt> <tt class="py-param">max_iterations</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L376"></a><tt class="py-lineno">376</tt>  <tt class="py-line">        <tt class="py-name">x0</tt> <tt class="py-op">=</tt> <tt class="py-name">x</tt> </tt>
<a name="L377"></a><tt class="py-lineno">377</tt>  <tt class="py-line">        <tt class="py-keyword">for</tt> <tt class="py-name">i</tt> <tt class="py-keyword">in</tt> <tt class="py-name">range</tt><tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">)</tt> <tt class="py-op">:</tt> </tt>
<a name="L378"></a><tt class="py-lineno">378</tt>  <tt class="py-line">            <tt class="py-name">x1</tt> <tt class="py-op">=</tt> <tt class="py-name">x0</tt> <tt class="py-op">-</tt> <tt class="py-name">f</tt><tt class="py-op">(</tt><tt class="py-name">x0</tt><tt class="py-op">)</tt><tt class="py-op">/</tt><tt class="py-name">fprime</tt><tt class="py-op">(</tt><tt class="py-name">x0</tt><tt class="py-op">)</tt> </tt>
<a name="L379"></a><tt class="py-lineno">379</tt>  <tt class="py-line">            <tt class="py-keyword">if</tt> <tt class="py-name">abs</tt><tt class="py-op">(</tt><tt class="py-name">x1</tt><tt class="py-op">-</tt><tt class="py-name">x0</tt><tt class="py-op">)</tt> <tt class="py-op">&lt;</tt> <tt class="py-name">tolerance</tt><tt class="py-op">:</tt> <tt class="py-keyword">return</tt> <tt class="py-name">x1</tt> </tt>
<a name="L380"></a><tt class="py-lineno">380</tt>  <tt class="py-line">            <tt class="py-name">x0</tt> <tt class="py-op">=</tt> <tt class="py-name">x1</tt> </tt>
<a name="L381"></a><tt class="py-lineno">381</tt>  <tt class="py-line">             </tt>
<a name="L382"></a><tt class="py-lineno">382</tt>  <tt class="py-line">        <tt class="py-keyword">raise</tt> <tt class="py-name">ArithmeticError</tt><tt class="py-op">(</tt> </tt>
<a name="L383"></a><tt class="py-lineno">383</tt>  <tt class="py-line">            <tt class="py-string">"Failed to converge after %d iterations, value is %f"</tt> \ </tt>
<a name="L384"></a><tt class="py-lineno">384</tt>  <tt class="py-line">                <tt class="py-op">%</tt> <tt class="py-op">(</tt><tt class="py-name">max_iterations</tt><tt class="py-op">,</tt><tt class="py-name">x1</tt><tt class="py-op">)</tt> <tt class="py-op">)</tt> </tt>
</div><a name="L385"></a><tt class="py-lineno">385</tt>  <tt class="py-line"> </tt>
<a name="L386"></a><tt class="py-lineno">386</tt>  <tt class="py-line">    <tt class="py-keyword">if</tt> <tt class="py-name">fprime</tt> <tt class="py-keyword">is</tt> <tt class="py-keyword">not</tt> <tt class="py-name">None</tt> <tt class="py-op">:</tt> </tt>
<a name="L387"></a><tt class="py-lineno">387</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">newton</tt><tt class="py-op">(</tt><tt class="py-name">f</tt><tt class="py-op">,</tt> <tt class="py-name">x</tt><tt class="py-op">,</tt> <tt class="py-name">fprime</tt><tt class="py-op">,</tt> <tt class="py-name">tolerance</tt><tt class="py-op">,</tt> <tt class="py-name">max_iterations</tt><tt class="py-op">)</tt> </tt>
<a name="L388"></a><tt class="py-lineno">388</tt>  <tt class="py-line">    <tt class="py-keyword">elif</tt> <tt class="py-name">y</tt> <tt class="py-keyword">is</tt> <tt class="py-keyword">not</tt> <tt class="py-name">None</tt> <tt class="py-op">:</tt> </tt>
<a name="L389"></a><tt class="py-lineno">389</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">bisect</tt><tt class="py-op">(</tt><tt class="py-name">f</tt><tt class="py-op">,</tt> <tt class="py-name">x</tt><tt class="py-op">,</tt> <tt class="py-name">y</tt><tt class="py-op">,</tt> <tt class="py-name">tolerance</tt><tt class="py-op">,</tt> <tt class="py-name">max_iterations</tt><tt class="py-op">)</tt> </tt>
<a name="L390"></a><tt class="py-lineno">390</tt>  <tt class="py-line">    <tt class="py-keyword">else</tt> <tt class="py-op">:</tt> </tt>
<a name="L391"></a><tt class="py-lineno">391</tt>  <tt class="py-line">        <tt class="py-keyword">return</tt> <tt class="py-name">secant</tt><tt class="py-op">(</tt><tt class="py-name">f</tt><tt class="py-op">,</tt> <tt class="py-name">x</tt><tt class="py-op">,</tt> <tt class="py-name">tolerance</tt><tt class="py-op">,</tt> <tt class="py-name">max_iterations</tt><tt class="py-op">)</tt> </tt>
</div><a name="L392"></a><tt class="py-lineno">392</tt>  <tt class="py-line"> </tt><script type="text/javascript">
<!--
expandto(location.href);
// -->
</script>
</pre>
<br />
<!-- ==================== NAVIGATION BAR ==================== -->
<table class="navbar" border="0" width="100%" cellpadding="0"
       bgcolor="#a0c0ff" cellspacing="0">
  <tr valign="middle">
  <!-- Home link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="weblogolib-module.html">Home</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Tree link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="module-tree.html">Trees</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Index link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="identifier-index.html">Indices</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Help link -->
      <th>&nbsp;&nbsp;&nbsp;<a
        href="help.html">Help</a>&nbsp;&nbsp;&nbsp;</th>

  <!-- Project homepage -->
      <th class="navbar" align="right" width="100%">
        <table border="0" cellpadding="0" cellspacing="0">
          <tr><th class="navbar" align="center"
            ><a class="navbar" target="_top" href="http://code.google.com/p/weblogo/">WebLogo</a></th>
          </tr></table></th>
  </tr>
</table>
<table border="0" cellpadding="0" cellspacing="0" width="100%%">
  <tr>
    <td align="left" class="footer">
    Generated by Epydoc 3.0.1 on Mon Jan 30 17:53:50 2012
    </td>
    <td align="right" class="footer">
      <a target="mainFrame" href="http://epydoc.sourceforge.net"
        >http://epydoc.sourceforge.net</a>
    </td>
  </tr>
</table>

<script type="text/javascript">
  <!--
  // Private objects are initially displayed (because if
  // javascript is turned off then we want them to be
  // visible); but by default, we want to hide them.  So hide
  // them unless we have a cookie that says to show them.
  checkCookie();
  // -->
</script>
</body>
</html>
